20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60/2.772
Lecture 18
Clausius Clapeyron equation
Condensed phase equilibria
• CLAUSIUS-CLAPERYON EQUATION
We derived the Clapeyron Equation last lecture:
ª S β − Sα º § ∆ S
§ dp ·
=«
¸
¨
» = ¨¨
© dT ¹ coexist ¬« V β − Vα »¼ © ∆V
·
¸¸
¹α → β
This is exact!
Let’s examine this for equilibrium between solid-liquid and solid-solid phases. For example, fusion:
∆S fus
§ dp ·
=
¨
¸
© dT ¹ coexist ∆Vfus
use
∆H fus = T ∆S fus
We can make some assumptions about these types of phase equilibria.
p2
Tm'
∆H fus dT
⋅
T
fus
Tm
³ dp = ³ ∆V
p1
p2 − p1 =
∆H fus
∆V fus
§ Tm' ·
ln¨¨ ¸¸
© Tm ¹
assume ∆Hfus, ∆Vfus are independent of p,T
Tm=melting temp @p2, Tm =melting temp @ p1
Tm’-Tm~0 (expect small change in Tm)
§ Tm' − Tm ·
§ Tm + Tm' − Tm ·
§ Tm' ·
¸¸
¸¸ = ln¨¨1 +
ln¨¨ ¸¸ = ln¨¨
T
T
T
m
m
¹
©
¹
©
© m¹
Since fraction is small, then
§ Tm' · § Tm' − Tm · ∆T
¸¸ =
ln¨¨ ¸¸ ≈ ¨¨
T
T
m
¹ T
© m¹ ©
Then
1
20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60/2.772
Lecture 18
∆H fus ∆Tm
⋅
∆Vfus Tm
∆p ≅
gives you ∆Tm, the melting point increase corresponding to a ∆p increase
also says: the coexistence line is approximately linear
• Equilibrium between gas/liquid or gas /solid (gas/condensed phase)
substance A:
A(l)= A(g)
or
A(s) = A(g)
∆Hvap
§ dp
¨
© dT
∆H
· ∆S
=
¸=
¹ ∆V T V g − V c
(
)
∆Hsub
Vc = Vsolid or Vliquid
a) We can approximate
V g − V c ≈ V g because Vg>>Vc
b) We can assume the gas is ideal
Vg =
RT
p
Clausius Clapeyron Equation
§ dp
¨
© dT
∆H
∆H ⋅ p
·
=
¸≈
RT 2
¹ T Vg
( )
§ d ln p · ∆ H
¸≈
¨
2
© dT ¹ RT
to get here we made approximations, but very good far from Tc.
Relates the vapor pressure of a liquid to ∆Hvap or vapor pressure of a solid to ∆Hsub.
c) We can make yet another approximation: assume ∆H independent of T. Then can integrate:
p
T
∆H
³ d ln p = ³ RT
p0
2
T0
2
dT
20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60/2.772
Lecture 18
§ p ·
∆H
ln ¨¨ ¸¸ ≈ −
R
© p0 ¹
§1 1 ·
¨¨ − ¸¸
© T T0 ¹
This equation allows us to get ∆Hsub from p
§ p·
∆H ∆H
ln ¨¨ ¸¸ ≈ −
+
p
RT
RT0
© 0¹
ln(p)
-∆H/R
0
3
1/T